How do I get the material's base color alpha work as transparency in the shader?
Base color value can be used as a driver of transparency. It is depending on shader if it can read and interpret that information. Principled shader is one which can. Diffuse shader won't be able to do it without mixing it with Transparent shader and driving Mix shader factor.

If you open color picker of Principled BSDF and set its alpha to 0 and color to white                  you will get only white material because shader renders it as basic diffuse surface by default.  

If you set Transmission value of the shader to 1 allowing it to happen shader will use alpha channel and will make your material as allowing light to come through it.  

However at this point it will refract light because by default shader's IOR value is set to 1.45 (common case of usage for interface between air and glass). Set that to 1.00 so no refraction happens and light comes straight.



Roughness value can be also taken down to 0 so it doesn't try to create rough surface. Object will still cast shadow of course; you will need to add more complex node setup or to disable shadow in material setitngs.